
Calculating \(A\)  via (REF ) naively requires computing each pseudoinverse \(n\)  times for a total of
\(\mathcal {O}(n^4)\)  operations.
As in §REF , this complexity can be lowered with an updating procedure:
following [1]}, we can efficiently compute \({R}_i^\dagger \)  using \({R}_{i+1}^\dagger \) .
Thus, we iterate through (REF ) backwards, starting at \(i=n\) .
when the (scalar) pseudoinverse is given by \({R}_{n,n}^\dagger = 1/{R}_{n,n}\) .
Proceeding inductively, we note that the next matrix block can be expressed as a rank-one update to the
previous block:
\({R}_{i} =\begin{bmatrix}0 & 0 \\0 & {R}_{i+1}\end{bmatrix}+ \begin{bmatrix}1 \\ 0\end{bmatrix}{R}_{i,i:n}.\) 
